Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

摘要

We determine the general structure of the partition function of the q-state Potts model in an external magnetic field, Z(G,q,v,w)for arbitrary q,temperature variable v,and magnetic field variable w,on cyclic,Mobius,and free strip graphs G of the square(sq),triangular (tri),and honeycomb(hc)lattices with width L-y and arbitrarily great length L-x.For the cyclic case we prove that the partition function has the form Z(Lambda,L-yxL-x,q,v,w)=Sigma(Ly)(d=0)(c)over tilde((d))Tr[(T-Z,T-(Lambda,Ly,)(d))(m)],where Lambda denotes the lattice type,(c)over tilde((d))are specified polynomials of degree d in q,T-Z,T(Lambda,Ly,)(d)is the corresponding transfer matrix,and m=L-x(L-x/2)for Lambda=sq,tri(hc),respectively. An analogous formula is given for Mobius strips,while only T-Z,T-(Lambda,Ly,)(d=0)appears for free strips.We exhibit a method for calculating T-Z,T-(Lambda, Ly,)(d)for arbitrary L-y and give illustrative examples.Explicit results for arbitrary L-y are presented for T-Z,T-(Lambda,Ly,)(d) with d=L-y and d=L-y-1.We find very simple formulas for the determinant det(T-Z,T-(Lambda,Ly,)(d)).We also give results for self-dual cyclic strips of the square lattice.